What is the optimal time complexity of solving Longest Bitonic Subsequence?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Longest Bitonic Subsequence?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Longest Bitonic Subsequence?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Longest Bitonic Subsequence?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingEasy

Longest Bitonic Subsequence

Learn how to solve the 'Longest Bitonic Subsequence' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function lbs(arr) that returns the length of the longest bitonic subsequence in an array arr. A subsequence is bitonic if it first increases and then decreases, or strictly increases, or strictly decreases.

Constraints

•1 <= len(arr) <= 1000
•-10^4 <= arr[i] <= 10^4

Examples

Example 1

Input

lbs([1, 11, 2, 10, 4, 5, 2, 1])

Output

Explanation

The longest bitonic subsequence is [1, 2, 10, 5, 2, 1] or [1, 2, 4, 5, 2, 1] with length 6.

Example 2

Input

lbs([12, 11, 40, 5, 3, 1])

Output

Explanation

The longest bitonic subsequence is [12, 11, 5, 3, 1] or [11, 40, 5, 3, 1] with length 5.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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